Long Time Behaviour of the Discrete Volume Preserving Mean Curvature Flow in the Flat Torus

TL;DR

This paper proves that the discrete volume preserving mean curvature flow in a flat torus converges exponentially fast to a translated stable shape, with a detailed analysis in two dimensions.

Contribution

It establishes long-term convergence results for the discrete flow near stable sets and provides a new Alexandrov-type estimate for periodic stable hypersurfaces.

Findings

Exponential convergence of the flow to a translate of the stable set.

New quantitative Alexandrov-type estimate for periodic stable hypersurfaces.

Complete characterization of long-term behavior in two dimensions.

Abstract

We show that the discrete approximate volume preserving mean curvature flow in the flat torus $\mathbb{T}^N$ starting near a strictly stable critical set $E$ of the perimeter converges in the long time to a translate of $E$ exponentially fast. As an intermediate result we establish a new quantitative estimate of Alexandrov type for periodic strictly stable constant mean curvature hypersurfaces. Finally, in the two dimensional case a complete characterization of the long time behaviour of the discrete flow with arbitrary initial sets of finite perimeter is provided.